Abstract

The asymptotically nonexpansive mappings have been introduced by Goebel and Kirkin 1972. Since then, a large number of authors have studied the weak and strongconvergence problems of the iterative algorithms for such a class of mappings.It is well known that the asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. In thepresent paper, we devote our study to the iterative algorithms for finding thefixed points of asymptotically pseudocontractive mappings in Hilbert spaces. Wesuggest an iterative algorithm and prove that it converges strongly to the fixedpoints of asymptotically pseudocontractive mappings.

MSC: 47J25, 47H09, 65J15.

Keywords

1 Introduction

Let H be a real Hilbert space with inner product 〈⋅,⋅〉 and norm ∥⋅∥, respectively. Let C be a nonempty, closed,and convex subset of H. Let T:C→C be a nonlinear mapping. We useF(T) to denote the fixed point set of T.

Recall that T is said to be L-Lipschitzian if there existsL>0 such that

∥Tx−Ty∥≤L∥x−y∥

for all x,y∈C. In this case, if L<1, then we call T anL-contraction. If L=1, we call Tnonexpansive. T is said to be asymptotically nonexpansiveif there exists a sequence {kn}⊂[1,∞) with limn→∞kn=1 such that

∥Tnx−Tny∥≤kn∥x−y∥

(1.1)

for all x,y∈C and all n≥1.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that, if C is a nonempty bounded, closed,and convex subset of a uniformly convex Banach space E, then everyasymptotically nonexpansive self-mapping T of C has a fixed point.Further, the set F(T) of fixed points of T is closed and convex.

Since then, a large number of authors have studied the following algorithms for theiterative approximation of fixed points of asymptotically nonexpansive mappings(see, e.g., [2–29] and the references therein).

An important class of asymptotically pseudocontractive mappings generalizing theclass of asymptotically nonexpansive mapping has been introduced and studied by Schuin 1991; see [19].

Recall that T:C→C is called an asymptotically pseudocontractivemapping if there exists a sequence {kn}⊂[1,∞) with limn→∞kn=1 for which the following inequality holds:

〈Tnx−Tny,x−y〉≤kn∥x−y∥2

(1.5)

for all x,y∈C and all n≥1. It is clear that (1.5) is equivalent to

∥Tnx−Tny∥2≤kn∥x−y∥2+∥(x−Tnx)−(y−Tny)∥2

(1.6)

for all x,y∈C and all n≥1.

Recall also that T is called uniformlyL-Lipschitzian if there exists L>0 such that

∥Tnx−Tny∥≤L∥x−y∥

for all x,y∈C and all n≥1.

Now, we know that the class of asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. If we define amapping T:[0,1]→[0,1] by the formula Tx=(1−x23)32, then we can verify that T is asymptoticallypseudocontractive but it is not asymptotically nonexpansive.

In order to approximate the fixed point of asymptotically pseudocontractive mappings,the following two results are interesting.

One is due to Schu [19], who proved the following convergence theorem.

Another one is due to Chidume and Zegeye [30] who introduced the following algorithm in 2003.

Let a sequence {xn} be generated from x1∈C by

xn+1=λnθnx1+(1−λn−λnθn)xn+λnTnxn,∀n≥1,

(1.7)

where the sequences {λn} and {θn} satisfy

(i)

∑n=1∞λnθn=∞ and λn(1+θn)≤1;

(ii)

λnθn→0, θn→0 and (θn−1θn−1)λnθn→0;

(iii)

kn−kn−1λnθn2→0;

(iv)

kn−1θn→0.

They gave the strong convergence analysis for the above algorithm (1.7) with somefurther assumptions on the mapping T in Banach spaces.

Remark 1.2 Note that there are some additional assumptions imposed on theunderlying space C and the mapping T in the above two results. In(1.7), the parameter control is also restricted.

Inspired by the results above, the main purpose of this article is to construct aniterative method for finding the fixed points of asymptotically pseudocontractivemappings. We construct an algorithm which is based on the algorithms (1.2) and(1.7). Under some mild conditions, we prove that the suggested algorithm convergesstrongly to the fixed point of asymptotically pseudocontractive mappingT.

2 Preliminaries

It is well known that in a real Hilbert space H, the following inequalityand equality hold:

Let{rn}be a sequence of real numbers. Assume{rn}does not decrease at infinity, that is, there exists at leasta subsequence{rnk}of{rn}such thatrnk≤rnk+1for allk≥0. For everyn≥N, define an integer sequence{τ(n)}as

3 Main results

Now we introduce the following iterative algorithm for asymptoticallypseudocontractive mappings.

Let C be a nonempty, closed, and convex subset of a real Hilbert spaceH. Let T:C→C be a uniformly L-Lipschitzian asymptoticallypseudocontractive mapping satisfying ∑n=1∞(kn−1)<∞. Let f:C→C be a ρ-contractive mapping. Let{αn}, {βn}, and {γn} be three real number sequences in[0,1].

Since the class of asymptotically nonexpansive mappings is a proper subclass of theclass of asymptotically pseudocontractive mappings and asymptotically nonexpansivemapping T is L-Lipschitzian with L=supnkn. Thus, from Theorem 3.2, we get the followingcorollary.

Then the sequence{xn}defined by (3.1) converges strongly tou=PF(T)f(u), which is the unique solution of the variationalinequality〈(I−f)x∗,x−x∗〉≥0for allx∈F(T).

Remark 3.4 Our Theorem 3.2 does not impose any boundedness or compactnessassumption on the space C or the mapping T. The parameter controlconditions (i)-(iii) are mild.

Remark 3.5 Our Corollary 3.3 is also a new result.

4 Conclusion

This work contains our dedicated study to develop and improve iterative algorithmsfor finding the fixed points of asymptotically pseudocontractive mappings in Hilbertspaces. We introduced our iterative algorithm for this class of problems, and wehave proven its strong convergence. This study is motivated by relevant applicationsfor solving classes of real-world problems, which give rise to mathematical modelsin the sphere of nonlinear analysis.

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.